https://r2.erweima.ai/mathbot/image/172618569247765ueis4j.jpg

The image shows a right triangle where:

- Angle \( A = 40^\circ \)
- The hypotenuse \( c \) is unknown.
- The adjacent side \( b \) is unknown.
- The opposite side to \( 50^\circ \) is given as 21 units.

Using the information from the triangle, we can apply trigonometric functions to find the unknown sides \( b \) and \( c \). Since the angle opposite to side \( b \) is \( 50^\circ \), we can use the following relations:

### 1. To find the hypotenuse \( c \) (using the sine function):
\[
\sin(50^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{21}{c}
\]
\[
c = \frac{21}{\sin(50^\circ)}
\]

### 2. To find side \( b \) (using the tangent function):
\[
\tan(50^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{b}
\]
\[
b = \frac{21}{\tan(50^\circ)}
\]

Let me calculate these values.The calculations give the following results:

- \( b \approx 17.62 \) units
- \( c \approx 27.41 \) units

Let me know if you want more details or have any questions!

Here are 5 related questions to consider:
1. How can we verify these results using the Pythagorean theorem?
2. What is the significance of the sine and tangent functions in right triangles?
3. How would the results change if angle \( A \) were different?
4. Can you calculate the area of this triangle with the given sides?
5. What would be the length of side \( b \) if the opposite side were longer?

**Tip:** Always double-check your angle measures when applying trigonometric functions to ensure accuracy.

Given a right triangle with angle A = 40°, angle B = 50°, and the opposite side of angle B is 21 units, find the hypotenuse and adjacent side.

Learn how to solve a right triangle where angle A is 40°, the opposite side is 21 units, and you need to find the adjacent side and hypotenuse. Using trigonometric functions like sine and tangent, this problem demonstrates key concepts in right triangle trigonometry, suitable for Grades 9-12.

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